The basic problem is optimizing the distribution of shifts between 
separate departments. 

A company has a number of call-service centers or departments $D$, and a 
fixed number of distinct operator skills or types $T$. A given demand 
specifies how many operators of each type are required at any given 
time, based on experience from previous years. The task is to distribute 
this demand across the departments in a way that best satisfies various 
requirements outlined in the next section. There is a finite number of 
distinct departments $D$, i.e. $|D|$ and there is a finite number of 
distinct shift types $T$, i.e. $|T|$. There is a known demand for each 
type is know for any given half hour interval in the phone service's 
open hours. 

\subsection{Distribution preferences} 

There is a number of preferences for the distribution. The most central 
requirement is the following:\\ 

\textsl{On average, each department should have a ratio of the weekly 
and quarterly shifts matching their ratio of the total call-center 
employees. } \\ 

This feature is central in making the shift distribution fair and 
acceptable for all departments. It entails a weighted distribution, 
where each department has been given a weight. This is to ensure that 
each department is not overloaded according to their respective 
capacity. Another central concern is \\ 

\textsl{All types of shifts need to be distributed equally among 
departments.}\\ 

Since each department should have competences in all fields they must be 
utilized by an equal distribution of all types of shifts. Finally there 
can be many specific constraints to the solution, like limits to the 
number of employees that can be working in a department at one time and 
days where some departments might be unavailable.\\ 

\textsl{Department shift duration preferences.}\\

The departments are not interested in having single half-hour shifts, surrounded
by empty shifts, and prefers longer shift duration. To discretize the problem
we assume that departments prefer to have full work days of shifts.
